A297925 Even numbers k such that k - 5 is prime but k - 3 is not prime.

12, 18, 24, 28, 36, 42, 48, 52, 58, 66, 72, 78, 84, 88, 94, 102, 108, 114, 118, 132, 136, 144, 156, 162, 168, 172, 178, 186, 198, 204, 216, 228, 234, 238, 246, 256, 262, 268, 276, 282, 288, 298, 312, 318, 322, 336, 342, 354, 358, 364, 372, 378, 384, 388, 394, 402, 406, 414, 426, 438, 444, 448, 454

Offset

1,1

Comments

Even numbers that are the sum of 5 and another prime, but not the sum of 3 and another prime. For n >= 1, a(n) - 5 = A049591(n), a(n) - 3 = A107986(n+1).

Let r(n) = a(n) - 5, Then r(n) is the greatest prime < a(n), and therefore A056240(a(n)) = 5*r(n). Furthermore, since r(n) + 2 must be composite, A056240(a(n)) = 5*A049591(n).

The terms in this sequence, combined with those in A298366 and A298252 form a partition of A005843(n);n>=3 (nonnegative even numbers>=6). This is because any even integer n>=6 satisfies either (i) n-3 is prime, (ii) n-5 is prime but n-3 is composite, or (iii) both n-5 and n-3 are composite.

Links

G. C. Greubel, Table of n, a(n) for n = 1..10000

Formula

a(n) = A049591(n) + 5 = A107986(n+1) + 3 for all n >= 1.

Example

12 is a term because 12 - 5 = 7 is prime, and 12 - 3 = 9 is composite. Also A049591(1)+5=7+5=12 and A107986(2)+3=9+3=12.
18 is a term because 18 - 5 = 13 is prime, and 18 - 3 = 15 is composite.
16 is not a term because 16 - 5 = 11 and 16 - 3 = 13 are both prime.

Maple

N:=100
for n from 8 to N by 2 do
if isprime(n-5) and not isprime(n-3) then print (n);
end if
end do

Mathematica

Select[Range[6, 500, 2], And[PrimeQ[# - 5], ! PrimeQ[# - 3]] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Range[6, 500, 2], Boole[PrimeQ[# -{5, 3}]] == {1, 0} &] (* Harvey P. Dale, Jan 30 2024 *)

Prog

(PARI) isok(n) = !(n % 2) && isprime(n-5) && !isprime(n-3); \\ Michel Marcus, Jan 09 2018
(Magma) [n: n in [3..500] | IsPrime(n-5) and not IsPrime(n-3) and (n mod 2) eq 0]; // G. C. Greubel, May 21 2019
(Sage) [n for n in (3..500) if is_prime(n-5) and not is_prime(n-3) and (mod(n, 2)==0)] # G. C. Greubel, May 21 2019
(GAP) Filtered([8..500], k-> IsPrime(k-5) and not IsPrime(k-3) and (n mod 2)=0) # G. C. Greubel, May 21 2019

Crossrefs

Similar to A130038. Subsequence of A175222.

Cf. A049591, A107986.

Sequence in context: A359929 A162151 A056773 * A341099 A175837 A136446

Adjacent sequences: A297922 A297923 A297924 * A297926 A297927 A297928

Keyword

nonn

Author

David James Sycamore, Jan 08 2018

Status

approved